Question

Dear Mitch,

Is there some way you can help me understand why it is that when you divide a regular, whole number by a fraction, the answer is bigger than the number you're dividing up? (I always thought that when you do division, the answer is smaller than the number you started with, especially since when you do multiplication, which is supposed to be the opposite of division, it makes things bigger. I still don't get it, even though I'm definitely supposed to get it by now (I'm only two years away from starting high school!)

Please help,

Jodie

Girl with fraction issues

Answer

Dear Jodie,

Before we answer your question, you need to click on...

Click it, and it'll be on its own window, and then press print. You can then close the new window, and you'll back here for the answer.

THEN cut it out with scissors. Set the scissors down nearby, because you're going to use them again in a moment. Okay, now. There is something I want you to read to yourself three times and read aloud a fourth time: 'Every time I hear the words 'divided by', and every time I see the division sign (two dots separated by a small horizontal), I will remind myself that the words 'divided by' is a shortened version of the full idea, which is: "...divided up into pieces of..." So if someone says 'twelve divided by six', you say to yourself: 'Twelve divided up into pieces of six.'

Now look at the paper ruler you've cut out. (And if you didn't get one off the screen, then find an old tape measure that no one is using, and look at it. Look at the twelve inches. And now say twelve inches divided up into pieces of wood that are six inches each. Okay, take a look. How many six inch pieces do you see?

Two?

Yes! That's what I got!

Oh.

So now, if you do the same thing with a fraction such as 1/3, it would be 12 inches of wood divided up into pieces of wood one-third inch each. Now you're thinking: Yeah, now it's harder. But it's not. Just do it one inch at a time. Look at one inch. How many pieces of wood that are one-third inch each do you think you'll be able to cut out of that first inch? That's right, and how many do you think you'll be able to cut out of the next inch of wood (or paper)? Right, another three. Every inch of wood or paper that you are dividing up you'll be able to divide into three. How many inches did we say we were dividing up into pieces? We said twelve inches. So if we get three out of every inch, then you can count 3-6-9-12-15-18-21.... And when you go all the way to twelve of your fingers counting off in threes, you should have 36 pieces, 12 inches of wood divided up into pieces of wood that are one-third-inch each will give you three each inch for twelve inches, or 36 altogether.

Hope it helps!

-- Mitch